Our procedure explicitly tests for stability against local distortions of periodicities up to \(2 \times 2\)  and thus provides a necessary, but not sufficient condition for dynamic stability. We stress, however, that even in cases where a material would spontaneously relax into a structure with periodicity larger than \(2 \times 2\) , the \(\Gamma \) -point dynamical matrix of the \(2\times 2\)  cell could exhibit negative eigenvalues. Our test is thus more stringent than it might seem at first glance. In principle, a rigorous test for dynamic stability would require the calculation of the full phonon band structure. Mathematically, the instabilities missed by our approach are those that result in imaginary phonons in the interior of the BZ but not at the zone boundary. Physically, such modes could be out of plane buckling or charge density wave-driven reconstructions with periodicities of several unit cells. In general, these types of instabilities are typically rather weak (as measured by the magnitude of the imaginary frequency) as compared to more local distortions such as the T to T\(^{\prime }\)  distortion considered below. Moreover, they could well be a special property of the isolated monolayer and become stabilised by the ubiquitous interactions of the 2D material with its environment, e.g. substrates. This is in fact supported by the full phonon calculations by Mounet et al. for \(\sim 250\)  isolated monolayers predicted to be easily exfoliable from experimentally known layered bulk phases[1]}. Indeed, most of the instabilities revealed by their calculations are of the type described above and would thus be missed by our test. However, these instabilities cannot be too critical as the monolayers are known to be stable in the vdW bonded layered bulk structure.
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